Van Emden HensonPanayot Vassilevski
Center for Applied Scientific Computing
Lawrence Livermore National Laboratory
Element-Free AMGe: General algorithms for computing
interpolation weights in AMG
International Workshop on Algebraic Multigrid
Sankt Wolfgang, AustriaJune 27, 2000
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Overview
AMG and AMGe
Element-Free AMGe: an interpolation rule based on neighborhood extensions
Examples of extensions: — -extension, — -extension— extensions from minimizing quadratic
functionals
Numerical experiments
AL2
This work was performed under the auspices of the U. S. Department of Energy by the University of California Lawrence Livermore National Laboratory under contract number: W-7405-Eng-48. UCRL-VG-138290
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AMG and AMGe
Assume a given sparse matrix AMG, or Algebraic Multigrid is MG based
only on the matrix entries. Essential components of AMG:
— a set, , of fine-grid degrees of freedom (dofs)
— a coarse grid, ; typically a subset of — a prolongation operator — smoothing iterations; typically Gauß-
Seidel or Jacobi— a coarse matrix given by
A
D
Dc DDDP c
PAPA Tc
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Coarse-grid selection
There are several ways to select the coarse grid
is typically a maximal independent set
each fine-grid dof is typically interpolated from a subset of its coarse nearest neighbors
Dc
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Building the prolongation, P
Let be a fine-grid dof Let be a neighborhood of Let be the coarse-grid dofs used to
interpolate a value at
Di Dic i
i
i ii \ cA
AAA
cfff
c i ii \ c
iD \
}}}
Examine rows of corresponding to .
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is the minimal neighborhood of Replace with modified version,
— by adding to all off-diagonal entries in i th row that are weakly connected to
— second, in all rows j for dofs strongly connected to i :– set
– set off-diagonals to zero
Prolongation in classical AMG
i iA ff A ff
a ii
aa jjjj cij cc
i
AA cfff 1
ith row of P is ith row of
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Key idea of AMGe: We can use information carried in the element stiffness matrices to determine— the nature of the smooth error components— accurate interpolation operators— the selection of the coarse grids
AMGe differs from standard AMG by using finite element information
Traditional AMG uses the following heuristic (based on M-matrices): smooth error varies slowest in the direction of “large” coefficients
New heuristic based on multigrid theory: interpolation must be able to reproduce a mode with error proportional to the size of the associated eigenvalue
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AMGe uses finite element stiffness matrices to localize the new heuristic
Local measure:
where are canonical basis vectors, and are sums of local stiffness matrices
If all local measures are small, the global measure is bounded and small good convergence!
Then solving a small local problem yields a row of the optimal interpolation (for the given set of C-points).
i Ai
eeA
eQIeQIM
i
Tii
Tii
ei
xam
0
BA kk
i i
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AMGe uses a small local problem to define prolongation
We can show that (for a given set of interpolation points), the “optimal” prolongation is the set of weights Q that satisfy the min/max problem:
Furthermore, solving the min/max problem is exactly equivalent to the following small matrix formulation
xamnim0Q
i
Tii
Tii
e
eeA
eQIeQI
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Prolongation in AMGe
The neighborhood is the union of the elements having i as vertex
We use the assembled neighborhood matrix
Partition the neighborhood matrix as before
ith row of P is ith row of
i
AA cfff 1
A i
iiAAA
c
ccfffi
\}
}
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Prolongation in AMGe, cont.
Note that there is no need to modify
Knowledge of the element matrices (used to create the assembled neighborhood matrix) carries with it implicitly the correct assignment and treatment of “weak” and “strong” connection. This is the main contribution of AMGe methods
AMGe produces superior prolongation. The goal of this work is to accomplish the superior prolongation without the knowledge of the element matrices
A ff
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AMGe - Richardson vs Gauß-Seidel
Two-level Multilevel
height amg amge amg amge
1 0.97 0.49 0.98 0.65
1/4 0.97 0.48 0.98 0.68
1/8 0.98 0.47 0.99 0.64
1/16 0.97 0.49 0.99 0.58
1/32 0.97 0.45 0.98 0.51
1/64 0.98 0.39 0.98 0.39
Richardson (1,0) cycle
Gauß-Seidel (1,0) cycle
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Prolongation in Element-free AMGe: based on extensions
Let be the f-point to which we wish to interpolate
i
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Prolongation in Element-free AMGe: based on extensions
Let be the f-point to which we wish to interpolate
is the set of points in the neighborhood of
i
i
i
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Prolongation in Element-free AMGe: based on extensions
Let be the f-point to which we wish to interpolate
is the set of points in the neighborhood of
is the set of coarse nearest neighbors of i
i
i
i
c i
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Prolongation in Element-free AMGe: based on extensions
Define , the set of “exterior” points for the neighborhood of : the set of points j such that j is connected to a fine point in the neighborhood of
X i
i
i
CkjX iijaiji \0
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Prolongation in Element-free AMGe: based on extensions
Define , the set of “exterior” points for the neighborhood of : the set of points j such that j is connected to a fine point in the neighborhood of
X i
i
i
CkjX iijaiji \0
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Prolongation in Element-free AMGe: based on extensions
We use the following window of the matrix A
where we will only be interested in the blocks shown.
X ic i
ii \ c
everything else on grid
AAA
AAA
XXcXfX
Xfcfff 0
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Prolongation in Element-free AMGe: based on extensions
Assume that an extension mapping is available:
i.e., we interpolate the exterior dofs (“X”) from the interior dofs f and c, by the rule
EEI
IE 0
0
cXfX
vEvEv ccXffXX
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Prolongation in Element-free AMGe: based on extensions
We construct the prolongation operator on the basis of the modified matrix
that is,
and
EEI
IAAAAA
cXfX
Xfcfffcfff 00
EAAA fXXfffff
EAAA cXXfcfcf
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Prolongation in Element-free AMGe: based on extensions
Then the ith row of the prolongation matrix P is taken as the ith row of the matrix
To make use of this method we must determine useful ways in which to build the extension operator
AA cfff 1
EEI
IE 0
0
cXfX
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Examples of extension: A-extension
Given defined on , we wish to extend it to defined on .
ivvX X i
Let be an exterior dof and define to be entries of A to which is connected.
iX
iX
ajS jiX 0
.The A-extension is:
vaa
v jjiSjiS
i XX
X
1
S iX
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Examples of extension: -extension
Given defined on , we wish to extend it to defined on .
ivvX X i
Let be an exterior dof and define to be entries of A to which is connected.
iX
iX
ajS jiX 0
The -extension is (a simple average):
S iX
L2
L2
vv jSS
iX
1
1
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Examples of extensions: based on minimizing a quadratic functional
This is the method Achi Brandt proposed in 1999.
Given defined on , find the extension by finding
where
This is a “simultaneous” extension, and is more expensive than -extension or -extension
v i vX
vQnim X
ii Xx
vvavQ jijiX XX 2
ij
A L2
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Examples of extensions: minimizing a “cut-off” quadratic functional
Given defined on , find the extension that satisfies
where and is a diagonal matrix. A good choice is the vector
note that this is also a “simultaneous” extension (but less expensive than the “full” quadratic)
v i vX
nimi
T
vXvBAvB
vvv
v
X
cf
c
fcXfXXXX AAA 1
1
1
II
B
X
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Classical AMG viewed as extension
The classical (Ruge-Stüben) AMG corresponds to selecting
and defining an -extension by setting
iii c
A
vaa
v jiji
i jXX
X
1
if is weakly connected to , and settingiX i
if is strongly connected to iX i
vv iiX
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Properties of the extensions
All the extensions described ensure that if is constant in then the extension is the same constant in , an important property for second-order elliptic PDEs.
For systems of PDEs we use the same extension mappings, based on the blocks of A associated with a given physical variable. That is, the extension to a dof corresponding to the physical variable k is based on dofs from that describe the same physical variable k.
v i
X ivX
iX i
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A simple example: the stretched quadrilateral
Consider the 2-D Laplacian operator, finite-element formulation on regular quadrialteral elements that are greatly stretched:
As , the operator stencil tends to:
hh yx »
141282141
h
h
y
x
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A simple example: the stretched quadrilateral: geometric approach
For this problem the standard geometric multigrid approach is to semicoarsen:
And the interpolation stencil is:
PA
0500
0500
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A simple example: the stretched quadrilateral: the AMG stencil
A simple calculation shows that classical AMG yields the interpolation stencil:
P GMA
380333380
380333380
211
31
211
211
31
211
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A simple example: the stretched quadrilateral: the neighborhood
Let
Define
then
N
S
NW
SW SE
NE
EW
EWi
ENESWNWSSNiiii c
A ff 8
A cf 111144
A Xf 22
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A simple example: the stretched quadrilateral: the A-extension
For the A-extension the extension operators are:
from which , and
yielding the interpolation stencil:
A cf 1111111131
A ff 21401
PA
830324830
830324830
621
6211
621
621
6211
621
E cX 4411
4411211 E fX
22
211
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A simple example: the stretched quadrilateral: the -extension
A similar calculation for the weights using the extension yields the interpolation stencil:
L2
L2
PL2
860463860
860463860
443
4461
443
443
4461
443
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A simple example: the stretched quadrilateral: the cut-off quadratic min
For the cut-off extension the extension operators are:
from which , and
yielding the interpolation stencil:
E fX 11
41E cX
44114411
81
A ff 7
A cf 00007721
PA
050
050
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The stencil produced by various extension methods:
Classical AMG
A-extension
L2-extension
cut-off quadratic
PA
830324830
830324830
P GMA
380333380
380333380
PL2
860463860
860463860
PA
050
050
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Numerical experiments
Second order elliptic operator
— Unstructured triangular mesh (400, 1600, 6400, and 25600 fine-grid elements)
— coarsened using agglomeration method of Jones & Vassilevski
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Element Agglomeration: Use graph theory to create coarse elements first, then select coarse-grid by abstracting geometric concepts of face, edge, vertex
AMGe requires coarse-grid elements & stiffness matrices.
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Coarsening history, unstructured 2nd order PDE
unstructured triangular grid
2nd order PDE
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Convergence results, unstructured 2nd order PDE
2nd order PDE problem— unstructured triangular grid— V(1,1) cycles— Gauß-Seidel smoothing
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Numerical experiments
2d elasticity (including thin-body)
— structured quadrilateral grid,
— is the beam thickness
— coarsened using agglomeration method
— the same coarse-grid is used for AMGe
(vertices of agglomerated elements)
hh yx
d 10
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Coarsening history, elasticity problem
Structured rectangular grid
2-d elasticity
thin body— d = 0.05
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Convergence results, elasticity
2d elasticity problem— d = 1— V(1,1) cycles— Gauß-Seidel smoothing
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Convergence results, elasticity
2d thin-body elasticity problem— d = 0.05— V(1,1) cycles— Gauß-Seidel smoothing
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Conclusions
AMGe produces superior prolongation, but requires extra information, i.e., the element matrices
For purely algebraic problems, element-free AMGe provides a technique for building pseudo-element matrices based on several extension schemes
Preliminary experimental results suggest that element-free AMGe also produces superior prolongation, competitive to AMGe results
Systems of PDEs can be handled in a very natural way
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Some papers of interest
Pre/Re-prints Available athttp://www.llnl.gov/CASC/linear_solvers
V. Henson and P. Vassilevski, “Element-free AMGe: General Algorithms for
computing Interpolation Weights in AMG,” submitted to SIAM Journal on Scientific
Computing.
M. Brezina, A. Cleary, R. Falgout, V. Henson, J. Jones, T. Manteuffel, S. McCormick, and J.
Ruge, “Algebraic Multigrid Based on Element Interpolation (AMGe),” to appear in
SIAM Journal on Scientific Computing.
J. Jones and P. Vassilevski, “AMGe Based on Element Agglomeration,” submitted to
SIAM Journal on Scientific Computing.
A. Cleary, R. Falgout, V. Henson, J. Jones, T. Manteuffel, S. McCormick, G. Miranda, and J.
Ruge, “Robustness and Scalability of Algebraic Multigrid,” SIAM Journal on Scientific
Computing, vol. 21 no. 5, pp. 1886-1908, 2000
A. Cleary, R. Falgout, V. Henson, J. Jones, “Coarse-Grid Selection for Parallel Algebraic
Multigrid,” in Proc. of the Fifth International Symposium on: Solving Irregularly Structured
Problems in Parallel, Lecture Notes in Computer Science, Springer-Verlag, New York, 1998.
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